Worry Wart

Webster's Ninth New Collegiate Dictionary dates it to 1936, but doesn't provide an origin.

I found this in the Google cache of a defunct Radiohead fan site:

"The OED claims the first usage of the phrase, spelled "worrywart," appeared on page 177 of Ivan Belknap's 1956 book Human Problems of State Mental Hospitals: "The persevering, nagging delusional group who were termed 'worry warts', 'nuisances', 'bird dogs', in the attendants' slang." Other reference sources claim the phrase first appeared in J. R. Williams's comic strip "Out Our Way" in 1956. "